Open App

JEE Exam > JEE Notes > Mathematics (Maths) for JEE Main & Advanced > Detailed Notes: Differential Equation

Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced PDF Download

Download, print and study this document offline

Please wait while the PDF view is loading

 Page 1


DIFFERENTIAL EQUATION 
Differential equations are a cornerstone of advanced mathematics, bridging the gap between 
calculus and real-world applications. These equations describe relationships involving functions 
and their derivatives, allowing us to model complex phenomena in physics, engineering, 
economics, and many other fields. 
This section introduces the fundamental concepts of differential equations, focusing on ordinary 
differential equations. We'll explore the definition, order, and degree of these equations, as well as 
methods for forming and solving them. By understanding differential equations, we gain powerful 
tools for analyzing dynamic systems and predicting how quantities change over time. 
From basic classifications to solution techniques, this material provides a foundation for working 
with differential equations, setting the stage for more advanced study and practical applications in 
various scientific disciplines. 
1. DIFFERENTIAL EQUATION : 
An equation that involves independent and dependent variables and the derivatives of the 
dependent variables is called a differential equation. 
A differential equation is said to be ordinary, if the differential coefficients have reference to a 
single independent variable only e.g. 
?? 2
?? ????
2
-
2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are 
two or more independent variables. e.g. 
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are 
concerned with ordinary differential equations only. 
2. ORDER OF DIFFERENTIAL EQUATION : 
The order of a differential equation is the order of the highest differential coefficient occurring in 
it. 
3. DEGREE OF DIFFERENTIAL EQUATION: 
The exponent of the highest order differential coefficient, when the differential equation is 
expressed as a polynomial in all the differential coefficient. 
Thus the differential equation : 
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ?? - 1
( ?? )
?? ?? ?? - 1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ?? 
Note : 
(i) The exponents of all the differential coefficient should be free from radicals and fraction. 
(ii) The degree is always positive natural number. 
(iii) The degree of differential equation may or may not exist. 
Page 2


DIFFERENTIAL EQUATION 
Differential equations are a cornerstone of advanced mathematics, bridging the gap between 
calculus and real-world applications. These equations describe relationships involving functions 
and their derivatives, allowing us to model complex phenomena in physics, engineering, 
economics, and many other fields. 
This section introduces the fundamental concepts of differential equations, focusing on ordinary 
differential equations. We'll explore the definition, order, and degree of these equations, as well as 
methods for forming and solving them. By understanding differential equations, we gain powerful 
tools for analyzing dynamic systems and predicting how quantities change over time. 
From basic classifications to solution techniques, this material provides a foundation for working 
with differential equations, setting the stage for more advanced study and practical applications in 
various scientific disciplines. 
1. DIFFERENTIAL EQUATION : 
An equation that involves independent and dependent variables and the derivatives of the 
dependent variables is called a differential equation. 
A differential equation is said to be ordinary, if the differential coefficients have reference to a 
single independent variable only e.g. 
?? 2
?? ????
2
-
2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are 
two or more independent variables. e.g. 
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are 
concerned with ordinary differential equations only. 
2. ORDER OF DIFFERENTIAL EQUATION : 
The order of a differential equation is the order of the highest differential coefficient occurring in 
it. 
3. DEGREE OF DIFFERENTIAL EQUATION: 
The exponent of the highest order differential coefficient, when the differential equation is 
expressed as a polynomial in all the differential coefficient. 
Thus the differential equation : 
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ?? - 1
( ?? )
?? ?? ?? - 1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ?? 
Note : 
(i) The exponents of all the differential coefficient should be free from radicals and fraction. 
(ii) The degree is always positive natural number. 
(iii) The degree of differential equation may or may not exist. 
Problem 1 : Find the order and degree of the following differential equation : 
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
 
(ii) 
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
) 
(iii) 
????
????
= v 3 ?? + 5 
Solution : 
(i) The given differential equation can be re-written as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
 
Hence order is 2 and degree is 3 . 
(ii) The given differential equation has the order 2. Since the given differential equation cannot be 
written as a polynomial in the differential coefficients, the degree of the equation is not defined. 
(iii) Its order is 1 and degree 1. 
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are - 
(A) 2,2 
(B) 2 , 3 
(C) 3,2 
(D) none of these 
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential 
equations). 
Ans. (A) 
4. FORMATION OF A DIFFERENTIAL 
EQUATION : 
In order to obtain a differential equation whose solution is 
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0 
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which 
we require ( ?? + 1 ) equations. 
A differential equation is obtained as follows : 
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the 
number of independent arbitrary constants in it. 
(b) Eliminate the arbitrary constants. 
(c) The eliminant is the required differential equation. 
Page 3


DIFFERENTIAL EQUATION 
Differential equations are a cornerstone of advanced mathematics, bridging the gap between 
calculus and real-world applications. These equations describe relationships involving functions 
and their derivatives, allowing us to model complex phenomena in physics, engineering, 
economics, and many other fields. 
This section introduces the fundamental concepts of differential equations, focusing on ordinary 
differential equations. We'll explore the definition, order, and degree of these equations, as well as 
methods for forming and solving them. By understanding differential equations, we gain powerful 
tools for analyzing dynamic systems and predicting how quantities change over time. 
From basic classifications to solution techniques, this material provides a foundation for working 
with differential equations, setting the stage for more advanced study and practical applications in 
various scientific disciplines. 
1. DIFFERENTIAL EQUATION : 
An equation that involves independent and dependent variables and the derivatives of the 
dependent variables is called a differential equation. 
A differential equation is said to be ordinary, if the differential coefficients have reference to a 
single independent variable only e.g. 
?? 2
?? ????
2
-
2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are 
two or more independent variables. e.g. 
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are 
concerned with ordinary differential equations only. 
2. ORDER OF DIFFERENTIAL EQUATION : 
The order of a differential equation is the order of the highest differential coefficient occurring in 
it. 
3. DEGREE OF DIFFERENTIAL EQUATION: 
The exponent of the highest order differential coefficient, when the differential equation is 
expressed as a polynomial in all the differential coefficient. 
Thus the differential equation : 
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ?? - 1
( ?? )
?? ?? ?? - 1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ?? 
Note : 
(i) The exponents of all the differential coefficient should be free from radicals and fraction. 
(ii) The degree is always positive natural number. 
(iii) The degree of differential equation may or may not exist. 
Problem 1 : Find the order and degree of the following differential equation : 
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
 
(ii) 
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
) 
(iii) 
????
????
= v 3 ?? + 5 
Solution : 
(i) The given differential equation can be re-written as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
 
Hence order is 2 and degree is 3 . 
(ii) The given differential equation has the order 2. Since the given differential equation cannot be 
written as a polynomial in the differential coefficients, the degree of the equation is not defined. 
(iii) Its order is 1 and degree 1. 
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are - 
(A) 2,2 
(B) 2 , 3 
(C) 3,2 
(D) none of these 
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential 
equations). 
Ans. (A) 
4. FORMATION OF A DIFFERENTIAL 
EQUATION : 
In order to obtain a differential equation whose solution is 
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0 
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which 
we require ( ?? + 1 ) equations. 
A differential equation is obtained as follows : 
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the 
number of independent arbitrary constants in it. 
(b) Eliminate the arbitrary constants. 
(c) The eliminant is the required differential equation. 
Note : 
(i) A differential equation represents a family of curves all satisfying some common properties. 
This can be considered as the geometrical interpretation of the differential equation. 
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h 
 order 
i.e. equal to number of independent arbitrary constant. 
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? -axis and 
having latus rectum ?? . 
Solution : Equation of parabola whose axes is parallel to ?? -axis and having latus rectum 'a' is ( ?? -
?? )
2
= ?? ( ?? - ?? ) 
Differentiating both sides, we get 2 ( ?? - ?? )
????
????
= ?? 
Again differentiating, we get 
? 2 ( ?? - ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0 
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
 
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
 
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
 
Writing the value of c from (i), we have 
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
 
i.e. 
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ?? 
Hence ?? =
2 ?? 3
[
????
????
] - ?? 
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
) - ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
, 
which is the required differential equation. 
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ?? 
?? ?? ?? ? ???? , where ?? = fixed constant 
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant 
Differentiating, we get 
????
????
= - ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ? 
Page 4


DIFFERENTIAL EQUATION 
Differential equations are a cornerstone of advanced mathematics, bridging the gap between 
calculus and real-world applications. These equations describe relationships involving functions 
and their derivatives, allowing us to model complex phenomena in physics, engineering, 
economics, and many other fields. 
This section introduces the fundamental concepts of differential equations, focusing on ordinary 
differential equations. We'll explore the definition, order, and degree of these equations, as well as 
methods for forming and solving them. By understanding differential equations, we gain powerful 
tools for analyzing dynamic systems and predicting how quantities change over time. 
From basic classifications to solution techniques, this material provides a foundation for working 
with differential equations, setting the stage for more advanced study and practical applications in 
various scientific disciplines. 
1. DIFFERENTIAL EQUATION : 
An equation that involves independent and dependent variables and the derivatives of the 
dependent variables is called a differential equation. 
A differential equation is said to be ordinary, if the differential coefficients have reference to a 
single independent variable only e.g. 
?? 2
?? ????
2
-
2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are 
two or more independent variables. e.g. 
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are 
concerned with ordinary differential equations only. 
2. ORDER OF DIFFERENTIAL EQUATION : 
The order of a differential equation is the order of the highest differential coefficient occurring in 
it. 
3. DEGREE OF DIFFERENTIAL EQUATION: 
The exponent of the highest order differential coefficient, when the differential equation is 
expressed as a polynomial in all the differential coefficient. 
Thus the differential equation : 
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ?? - 1
( ?? )
?? ?? ?? - 1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ?? 
Note : 
(i) The exponents of all the differential coefficient should be free from radicals and fraction. 
(ii) The degree is always positive natural number. 
(iii) The degree of differential equation may or may not exist. 
Problem 1 : Find the order and degree of the following differential equation : 
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
 
(ii) 
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
) 
(iii) 
????
????
= v 3 ?? + 5 
Solution : 
(i) The given differential equation can be re-written as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
 
Hence order is 2 and degree is 3 . 
(ii) The given differential equation has the order 2. Since the given differential equation cannot be 
written as a polynomial in the differential coefficients, the degree of the equation is not defined. 
(iii) Its order is 1 and degree 1. 
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are - 
(A) 2,2 
(B) 2 , 3 
(C) 3,2 
(D) none of these 
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential 
equations). 
Ans. (A) 
4. FORMATION OF A DIFFERENTIAL 
EQUATION : 
In order to obtain a differential equation whose solution is 
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0 
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which 
we require ( ?? + 1 ) equations. 
A differential equation is obtained as follows : 
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the 
number of independent arbitrary constants in it. 
(b) Eliminate the arbitrary constants. 
(c) The eliminant is the required differential equation. 
Note : 
(i) A differential equation represents a family of curves all satisfying some common properties. 
This can be considered as the geometrical interpretation of the differential equation. 
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h 
 order 
i.e. equal to number of independent arbitrary constant. 
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? -axis and 
having latus rectum ?? . 
Solution : Equation of parabola whose axes is parallel to ?? -axis and having latus rectum 'a' is ( ?? -
?? )
2
= ?? ( ?? - ?? ) 
Differentiating both sides, we get 2 ( ?? - ?? )
????
????
= ?? 
Again differentiating, we get 
? 2 ( ?? - ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0 
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
 
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
 
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
 
Writing the value of c from (i), we have 
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
 
i.e. 
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ?? 
Hence ?? =
2 ?? 3
[
????
????
] - ?? 
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
) - ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
, 
which is the required differential equation. 
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ?? 
?? ?? ?? ? ???? , where ?? = fixed constant 
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant 
Differentiating, we get 
????
????
= - ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ? 
Again differentiating, we get 
?? 2
?? ?? ?? 2
= - ?? 2
?? ?? ?? ?? ? ???? - ?? 2
?? ?? ?? ?? ? ???? 
using equation (i), we get ?
?? 2
?? ?? ?? 2
= - ?? 2
?? 
Do yourself - 2 
Eliminate the arbitrary constants and obtain the differential equation satisfied by it. 
(i) ?? = 2 ?? + ????
?? 
(ii) ?? = (
?? ?? 2
) + ???? 
(iii) ?? = ?? ?? 2 ?? + ?? ?? - 2 ?? + ?? 
5. SOLUTION OF DIFFERENTIAL 
EQUATION : 
The solution of the differential equation is a relation between the variables of the equation not 
containing the derivatives, but satisfying the given differential equation (i.e., from which the given 
differential equation can be derived). 
Thus, the solution of 
????
????
= ?? ?? could be obtained by simply integrating both sides, i.e., ?? = ?? ?? + ?? 
and that of, 
????
????
= ???? + ?? is ?? =
????
2
2
+ ???? + ?? , where ?? is arbitrary constant. 
(i) A general solution or an integral of a differential equation is a relation between the variables 
(not involving the derivatives) which contains the same number of the arbitrary constants as the 
order of the differential equation. 
For example, a general solution of the differential equation 
?? 2
?? ?? ?? 2
= - 4 ?? is ?? = ???? ?? ?? ? 2 ?? + ?? ?? ?? ?? ? 2 ?? 
where ?? and ?? are the arbitrary constants. 
(ii) Particular solution or particular integral is that solution of the differential equation which is 
obtained from the general solution by assigning particular values to the arbitrary constant in the 
general solution. 
For example, ?? = 10 ?? ?? ?? ? 2 ?? + 5 ?? ?? ?? ? 2 ?? is a particular solution of differential equation 
?? 2
?? ????
2
= - 4 ?? . 
Note : 
(i) The general solution of a differential equation can be expressed in different (but equivalent) 
forms. For example 
?????? ? ?? - ?????? ? ( ?? + 2 ) = ?? # ( ?? ) ? 
where ?? is an arbitrary constant is the general solution of the differential equation ????
'
= ?? + 2. 
The solution given by equation (i) can also be re-written as 
?????? ? (
?? ?? + 2
) = ?? ???? 
?? ?? + 2
= ?? ?? = ?? 1
# ( ???? ) ? ???? ? ?? = ?? 1
( ?? + 2 ) # ( ?????? ) ? 
where ?? 1
= ?? ?? is another arbitrary constant. The solution (iii) can also be written as 
Page 5


DIFFERENTIAL EQUATION 
Differential equations are a cornerstone of advanced mathematics, bridging the gap between 
calculus and real-world applications. These equations describe relationships involving functions 
and their derivatives, allowing us to model complex phenomena in physics, engineering, 
economics, and many other fields. 
This section introduces the fundamental concepts of differential equations, focusing on ordinary 
differential equations. We'll explore the definition, order, and degree of these equations, as well as 
methods for forming and solving them. By understanding differential equations, we gain powerful 
tools for analyzing dynamic systems and predicting how quantities change over time. 
From basic classifications to solution techniques, this material provides a foundation for working 
with differential equations, setting the stage for more advanced study and practical applications in 
various scientific disciplines. 
1. DIFFERENTIAL EQUATION : 
An equation that involves independent and dependent variables and the derivatives of the 
dependent variables is called a differential equation. 
A differential equation is said to be ordinary, if the differential coefficients have reference to a 
single independent variable only e.g. 
?? 2
?? ????
2
-
2 ????
????
+ ?? ?? ?? ? ?? = 0 and it is said to be partial if there are 
two or more independent variables. e.g. 
?? ?? ????
+
????
????
+
????
????
= 0 is a partial differential equation. We are 
concerned with ordinary differential equations only. 
2. ORDER OF DIFFERENTIAL EQUATION : 
The order of a differential equation is the order of the highest differential coefficient occurring in 
it. 
3. DEGREE OF DIFFERENTIAL EQUATION: 
The exponent of the highest order differential coefficient, when the differential equation is 
expressed as a polynomial in all the differential coefficient. 
Thus the differential equation : 
?? ( ?? , ?? ) [
?? ?? ?? ?? ?? ?? ]
?? + ?? ( ?? , ?? ) [
?? ?? - 1
( ?? )
?? ?? ?? - 1
]
?? + ? … . . . = 0 ???? ????? ? ?? ???? ?? ?? ? ?? ? \ & ? ?? ?? ?? ???? ?? ?? 
Note : 
(i) The exponents of all the differential coefficient should be free from radicals and fraction. 
(ii) The degree is always positive natural number. 
(iii) The degree of differential equation may or may not exist. 
Problem 1 : Find the order and degree of the following differential equation : 
(i) v
?? 2
?? ?? ?? 2
= v
????
????
+ 3
3
 
(ii) 
?? 2
?? ?? ?? 2
= ?? ?? ?? ? (
????
????
) 
(iii) 
????
????
= v 3 ?? + 5 
Solution : 
(i) The given differential equation can be re-written as (
?? 2
?? ????
2
)
3
= (
????
????
+ 3 )
2
 
Hence order is 2 and degree is 3 . 
(ii) The given differential equation has the order 2. Since the given differential equation cannot be 
written as a polynomial in the differential coefficients, the degree of the equation is not defined. 
(iii) Its order is 1 and degree 1. 
Problem 2: The order and degree of the differential equation (
?? 2
? ?? ????
2
)
2
+ 3 (
????
????
)
3
+ 4 = 0 are - 
(A) 2,2 
(B) 2 , 3 
(C) 3,2 
(D) none of these 
Solution : Clearly order is 2 and degree is 2 (from the definition of order and degree of differential 
equations). 
Ans. (A) 
4. FORMATION OF A DIFFERENTIAL 
EQUATION : 
In order to obtain a differential equation whose solution is 
?? ( ?? 1
, ?? 1
, ?? 1
, ?? 2
, ?? 3
… … … , ?? ?? ) = 0 
where ?? 1
, ?? 2
, … … . . ?? ?? are ' ?? ' arbitrary constants, we have to eliminate the ' ?? ' constants for which 
we require ( ?? + 1 ) equations. 
A differential equation is obtained as follows : 
(a) Differentiate the given equation w.r.t the independent variable (say ?? ) as many times as the 
number of independent arbitrary constants in it. 
(b) Eliminate the arbitrary constants. 
(c) The eliminant is the required differential equation. 
Note : 
(i) A differential equation represents a family of curves all satisfying some common properties. 
This can be considered as the geometrical interpretation of the differential equation. 
(ii) For there being ?? differentiation, the resulting equation must contain a derivative of ?? ?? h 
 order 
i.e. equal to number of independent arbitrary constant. 
Problem 3: Find the differential equation of all parabolas whose axes is parallel to the ?? -axis and 
having latus rectum ?? . 
Solution : Equation of parabola whose axes is parallel to ?? -axis and having latus rectum 'a' is ( ?? -
?? )
2
= ?? ( ?? - ?? ) 
Differentiating both sides, we get 2 ( ?? - ?? )
????
????
= ?? 
Again differentiating, we get 
? 2 ( ?? - ?? )
?? 2
?? ?? ?? 2
+ 2 (
????
????
)
2
= 0 ? ? ? ?? ?? 2
?? ?? ?? 2
+ 2 (
????
????
)
3
= 0 
Problem 4: Find the differential equation whose solution represents the family : ?? ( ?? + ?? )
2
= ?? 3
 
Solution : ? ?? ( ?? + ?? )
2
= ?? 3
 
Differentiating, we get, c. [ 2 ( ?? + ?? ) ]
????
????
= 3 ?? 2
 
Writing the value of c from (i), we have 
2 ?? 3
( ?? + ?? )
2
( ?? + ?? )
????
????
= 3 ?? 2
? ?
2 ?? 3
?? + ?? 2 ?? 3
?? + ?? = 3 ?? 2
 
i.e. 
2 ?? ?? + ?? ????
????
= 3 ?
2 ?? 3
[
????
????
] = ?? + ?? 
Hence ?? =
2 ?? 3
[
????
????
] - ?? 
Substituting value of ?? in equation (i), we get [
2 ?? 3
(
????
????
) - ?? ] [
2 ?? 3
????
????
]
2
= ?? 3
, 
which is the required differential equation. 
Problem 5 : Find the differential equation whose solution represents the family : ?? = ?? ?? ?? ?? ? ???? + ?? 
?? ?? ?? ? ???? , where ?? = fixed constant 
Solution : ? ?? = ?? ?? ?? ?? ? ???? + ?? ?? ?? ?? ? ???? , ?? = fixed constant 
Differentiating, we get 
????
????
= - ???? ?? ?? ?? ? ???? + ???? ?? ?? ?? ? ???? # ( ?? ) ? 
Again differentiating, we get 
?? 2
?? ?? ?? 2
= - ?? 2
?? ?? ?? ?? ? ???? - ?? 2
?? ?? ?? ?? ? ???? 
using equation (i), we get ?
?? 2
?? ?? ?? 2
= - ?? 2
?? 
Do yourself - 2 
Eliminate the arbitrary constants and obtain the differential equation satisfied by it. 
(i) ?? = 2 ?? + ????
?? 
(ii) ?? = (
?? ?? 2
) + ???? 
(iii) ?? = ?? ?? 2 ?? + ?? ?? - 2 ?? + ?? 
5. SOLUTION OF DIFFERENTIAL 
EQUATION : 
The solution of the differential equation is a relation between the variables of the equation not 
containing the derivatives, but satisfying the given differential equation (i.e., from which the given 
differential equation can be derived). 
Thus, the solution of 
????
????
= ?? ?? could be obtained by simply integrating both sides, i.e., ?? = ?? ?? + ?? 
and that of, 
????
????
= ???? + ?? is ?? =
????
2
2
+ ???? + ?? , where ?? is arbitrary constant. 
(i) A general solution or an integral of a differential equation is a relation between the variables 
(not involving the derivatives) which contains the same number of the arbitrary constants as the 
order of the differential equation. 
For example, a general solution of the differential equation 
?? 2
?? ?? ?? 2
= - 4 ?? is ?? = ???? ?? ?? ? 2 ?? + ?? ?? ?? ?? ? 2 ?? 
where ?? and ?? are the arbitrary constants. 
(ii) Particular solution or particular integral is that solution of the differential equation which is 
obtained from the general solution by assigning particular values to the arbitrary constant in the 
general solution. 
For example, ?? = 10 ?? ?? ?? ? 2 ?? + 5 ?? ?? ?? ? 2 ?? is a particular solution of differential equation 
?? 2
?? ????
2
= - 4 ?? . 
Note : 
(i) The general solution of a differential equation can be expressed in different (but equivalent) 
forms. For example 
?????? ? ?? - ?????? ? ( ?? + 2 ) = ?? # ( ?? ) ? 
where ?? is an arbitrary constant is the general solution of the differential equation ????
'
= ?? + 2. 
The solution given by equation (i) can also be re-written as 
?????? ? (
?? ?? + 2
) = ?? ???? 
?? ?? + 2
= ?? ?? = ?? 1
# ( ???? ) ? ???? ? ?? = ?? 1
( ?? + 2 ) # ( ?????? ) ? 
where ?? 1
= ?? ?? is another arbitrary constant. The solution (iii) can also be written as 
?? + 2 = ?? 2
?? 
where ?? 2
= 1 / ?? 1
 is another arbitrary constant. 
(ii) All differential equations that we come across have unique solutions or a family of solutions. 
For example, the differential equation |
????
????
| + | ?? | = 0 has only the trivial solution, i.e. ?? = 0. 
The differential equation |
????
????
| + | ?? | + ?? = 0 , ?? > 0 has no solution. 
6. ELEMENTARY TYPES OF FIRST ORDER & 
FIRST DEGREE DIFFERENTIAL EQUATIONS : 
(a) Separation of Variables : 
Some differential equations can be solved by the method of separation of variables (or "variable 
separable"). This method is only possible, if we can express the differential equation in the form 
?? ( ?? ) ???? + ?? ( ?? ) ???? = 0 
where ?? ( ?? ) is a function of ' ?? ' only and ?? ( ?? ) is a function of ' ?? ' only. 
A general solution of this is given by, 
? ?
?
?
? ?? ( ?? ) ???? + ? ?
?
?
? ?? ( ?? ) ???? = ?? 
where 'c' is the arbitrary constant. 
Problem 6 : Solve the differential equation ????
????
????
=
1 + ?? 2
1 + ?? 2
( 1 + ?? + ?? 2
). 
Solution : Differential equation can be rewritten as 
????
????
????
= ( 1 + ?? 2
) ( 1 +
?? 1 + ?? 2
) ?
?? 1 + ?? 2
???? = (
1
?? +
1
1 + ?? 2
) ???? 
Integrating, we get 
1
2
???? ? ( 1 + ?? 2
) = ?? ?? ?? + ?? ?? ?? - 1
? ?? + ?????? ? v 1 + ?? 2
= ???? ?? ?? ?? ?? - 1
? ?? . 
Ans. 
Problem 7: Solve the differential equation ( ?? 3
- ?? 2
?? 3
)
????
????
+ ?? 3
+ ?? 2
?? 3
= 0. 
Solution: ? The given equation ( ?? 3
- ?? 2
?? 3
)
????
????
+ ?? 3
+ ?? 2
?? 3
= 0 
Case-I : ?? = 0 is one solution of differential equation 
Case-II : If ?? ? 0, then

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|443 docs\|143 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

Join Course for Free

Top Courses for JEE

View all

FAQs on Detailed Notes: Differential Equation - Mathematics (Maths) for JEE Main & Advanced

1. What is a differential equation?

Ans. A differential equation is an equation involving an unknown function and its derivatives. It relates the rate of change of the function to the function itself.

2. How are differential equations used in real-life applications?

Ans. Differential equations are used in various fields such as physics, engineering, biology, and economics to model and analyze systems that change over time.

3. What is the order of a differential equation?

Ans. The order of a differential equation is the highest order of the derivative present in the equation. For example, a first-order differential equation contains the first derivative of the function.

4. How do you solve a differential equation?

Ans. Differential equations can be solved using various methods such as separation of variables, integrating factors, and substitution techniques. The solution typically involves finding the function that satisfies the given equation.

5. What is the importance of differential equations in the field of mathematics?

Ans. Differential equations play a crucial role in mathematics as they provide a powerful tool for modeling and analyzing complex systems. They are used to describe relationships between quantities that change continuously and are fundamental in various branches of science and engineering.

Related Exams

JEE

About this Document

	4.96/5 Rating
	Nov 23, 2024 Last updated

Document Description: Detailed Notes: Differential Equation for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The notes and questions for Detailed Notes: Differential Equation have been prepared according to the JEE exam syllabus. Information about Detailed Notes: Differential Equation covers topics like and Detailed Notes: Differential Equation Example, for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Detailed Notes: Differential Equation.

Introduction of Detailed Notes: Differential Equation in English is available as part of our Mathematics (Maths) for JEE Main & Advanced for JEE & Detailed Notes: Differential Equation in Hindi for Mathematics (Maths) for JEE Main & Advanced course. Download more important topics related with notes, lectures and mock test series for JEE Exam by signing up for free. JEE: Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced

Description

Full syllabus notes, lecture & questions for Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced - JEE | Plus excerises question with solution to help you revise complete syllabus for Mathematics (Maths) for JEE Main & Advanced | Best notes, free PDF download

Information about Detailed Notes: Differential Equation

In this doc you can find the meaning of Detailed Notes: Differential Equation defined & explained in the simplest way possible. Besides explaining types of Detailed Notes: Differential Equation theory, EduRev gives you an ample number of questions to practice Detailed Notes: Differential Equation tests, examples and also practice JEE tests

	Mathematics (Maths) for JEE Main & Advanced 209 videos\|443 docs\|143 tests

Mathematics (Maths) for JEE Main & Advanced

209 videos|443 docs|143 tests

Join Course for Free

Download as PDF

Explore Courses for JEE exam

Top Courses for JEE

Explore Courses

Signup for Free!

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.

Start learning for Free

10M+ students study on EduRev

Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced

Free

practice quizzes

ppt

shortcuts and tricks

Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced

Semester Notes

Viva Questions

past year papers

Objective type Questions

Extra Questions

Important questions

Summary

video lectures

pdf

study material

Previous Year Questions with Solutions

mock tests for examination

Detailed Notes: Differential Equation | Mathematics (Maths) for JEE Main & Advanced

Exam

MCQs

Sample Paper

;

Additional Information about Detailed Notes: Differential Equation for JEE Preparation

Detailed Notes: Differential Equation Free PDF Download

The Detailed Notes: Differential Equation is an invaluable resource that delves deep into the core of the JEE exam. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. With the help of these notes, you can grasp complex subjects quickly, revise important points easily, and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner, allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material, or toppers' notes, this PDF has got you covered. Download the Detailed Notes: Differential Equation now and kickstart your journey towards success in the JEE exam.

Importance of Detailed Notes: Differential Equation

The importance of Detailed Notes: Differential Equation cannot be overstated, especially for JEE aspirants. This document holds the key to success in the JEE exam. It offers a detailed understanding of the concept, providing invaluable insights into the topic. By knowing the concepts well in advance, students can plan their preparation effectively. Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.

Detailed Notes: Differential Equation

Detailed Notes: Differential Equation Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to Detailed Notes: Differential Equation. It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation. Practice papers and question papers enable you to assess your progress effectively. Additionally, the paper analysis provides valuable tips for tackling the exam strategically. Access to Toppers' notes gives you an edge in understanding complex concepts. Whether you're a beginner or aiming for advanced proficiency, Detailed Notes: Differential Equation Notes on EduRev are your ultimate resource for success.

Detailed Notes: Differential Equation JEE Questions

The "Detailed Notes: Differential Equation JEE Questions" guide is a valuable resource for all aspiring students preparing for the JEE exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

Study Detailed Notes: Differential Equation on the App

Students of JEE can study Detailed Notes: Differential Equation alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the Detailed Notes: Differential Equation, students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc. The EduRev App will make your learning easier as you can access it from anywhere you want. The content of Detailed Notes: Differential Equation is prepared as per the latest JEE syllabus.

Education Revolution